Can I state the Mandelbrot Set/Equation like this?

But it must be under iteration. Then definitely you can do that.
Here I suppose you are saying z and c are complex numbers.

 

Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

 

??? Just "iterating a function" will give you a lot of numbers and a lot of points where the iteration does not converge to any number. Again, the "Mandlebrot set" is the set of complex numbers for which the iteration does converge.

I don't know what you mean by "summarized" here. What do you do with that iteration to get the Mandlebrot set. (The "Julia sets" use that same iteration but in a different way.)

 

Given f(z) = z² + c then z_n+1 = f(z_n). z₀=0, c ranges.

The mandelbrot set is over the domain, c.

If you want to produce a mandelbrot set, you need more than an equations. You need an algorithm.

 

It all depends on context. If you are talking to somebody about dynamical systems, then yes, you can write it like that. If you're not in the correct context, then no, you can't. It all depends on whether the other person will understand you or not.